6. Normal Distribution and Continuous Random Variables

Which of the following is a property of the standard normal distribution?

Which of the following data sets best approximates a standard normal distribution?

Given the mean of a normal distribution is $\mu$ and the standard deviation is $\sigma$, what is the mean of the corresponding standard normal distribution after standardizing the variable?

For the standard normal distribution, what is the area to the right of $1$ (that is, $P(Z>1)$)?

Using the standard normal distribution, what is the value of $P(0<z<2.25)$?

Suppose the random variable $Z$ follows a standard normal distribution. What is the area under the standard normal curve to the left of $Z=1$?

In a standard normal distribution, what percentage of the data falls within one standard deviation of the mean (that is, between $-1$ and $1$ on the $z$-score scale)?

In the $standardnormal$ distribution, what happens to the graph of the normal curve as the mean $\mu$ increases while the standard deviation $\sigma$ remains constant?

Assume $x$ is normally distributed with a mean of $10$ and a standard deviation of $2$. What is the z-score for $x=14$?

For a standard normal distribution, which of the following variables always equals $0$?

Which of the following best describes a key difference between the $t$-distribution and the standard normal ($z$) distribution?

Car Colors

In Exercises 9–12, assume that 100 cars are randomly selected. Refer to the accompanying graph, which shows the top car colors and the percentages of cars with those colors (based on PPG Industries).

Black Cars Find the probability that at least 25 cars are black. Is 25 a significantly high number of black cars?

For the standard normal distribution, which of the following is equivalent to $p(z\ge 1.4)$?

Water Taxi Safety When a water taxi sank in Baltimore’s Inner Harbor, an investigation revealed that the safe passenger load for the water taxi was 3500 lb. It was also noted that the mean weight of a passenger was assumed to be 140 lb. Assume a “worst-case” scenario in which all of the passengers are adult men. Assume that weights of men are normally distributed with a mean of 188.6 lb and a standard deviation of 38.9 lb (based on Data Set 1 “Body Data” in Appendix B).

c. With a load limit of 3500 lb, how many male passengers are allowed if we assume the updated mean weight of 188.6 lb?

Which of the following is a characteristic of the standard normal distribution?